Other Ensembles

As mentioned above, fuUy grand-canonical ensembles are difficult to use for polymers, due to the problem of inserting chains in a random, unbiased configuration in a dense polymer melt. But of course, the standard canonical 

One can obtain information on the phase behavior in equilibrium from studies such as these in several ways one approach rests on generalization of techniques for the estimation of chemical potentials in dense polymer 

Another technique rests on calculating the structure factor S q). As demonstrated first by Sariban and Binder, though in the framework of a semi-grand-canonical simulation, one can estimate the spinodal curve from a linear extrapolation of versus e/ksT estimating the 

A rather interesting method, motivated by successful applications to studies of liquid-gas phase separation of Lennard-Jones fluids and other fluids in the canonical ensemble, starts from the observation that the temperature variation of the order parameter distribution can be analyzed in subsystems of linear dimension 1. Although in the total system (of linear dimension L) the order parameter is conserved, for CL neighboring subsystems can freely exchange different types of monomers (and chains). Of course, subsystems must be large enough, so that they still contain many chains. 

5 apply for angles tt, 27t/3, jr/2 and 7t/3 between successive bonds. Here temperature is measured in energy units. (From Gauger and Pakula. ) 

The discussion so far has focused on the use of canonical ensemble averages like (2), in which the variables (N, V, T) are held fixed. That is the most common application of MC work to fluids. However, averages in any ensemble can be put into form (1) and evaluated using similar techniques. In practice computations have been carried out in the grand canonical ensemble [with (fi, V, T) fixed] and in the isothermal isobaric [with (N, P, 7) fixed]. In the former the number of particles N is a fluctuating quantity that must be sampled during the MC experiment, while in the latter this is true of the volume V. 

One requires the configurational probability densities for the states of the two ensembles [cf. Eq. (3)]. For the grand canonical ensemble, this is given by 

The (N, P, T) ensemble will sometimes have advantages over the N, V, T). Evidently in the latter the values of V and T necessary to give a certain pressure are not known in advance, and the result can be far from the conditions of interest. If one wants to compare results at a common pressure, or to compare them with experimental results at fixed pressure, it may often be sensible to fix the pressure and use the (N, P, T) ensemble. The equation of state, in the form (V(N, P, T)), is measured rather more directly in the (N, P, T) ensemble and may sometimes be more precise. This possible advantage can certainly be realized for hard-core particles, where the (N, V, T) pressure determination requires an often dubious extrapolation of g2 to the contact distance of the hard cores. For other thermodynamic quantities, such as the energy, the (N, P, T) method seems to be marginally less economical. 

Another area that has attracted some attention is the problem of polymer configurations. Early work on polymers was concerned with analytic investigation of random walk models in which monomers, far apart along the contour of 

LaP has explored an alternative idea in which a step (i) is chosen uniformly from 1,2. n and the set of vectors l,+i, l,+2. is replaced by corresponding vectors obtained by rotation about the direction determined by 1,. This approach has been applied to polymers with longer range forces. 

In all the previous sections we have used the definition of the chemical potential in the r, K, N ensemble (Eq. 5.9.1). This was done mainly for convenience. In actual applications, and in particular when comparison with experimental results is required, it is necessary to use the T, P, N ensemble. In that case, the chemical potential is defined by 

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Since other sets of constraints can be used, there are other ensembles and other partition functions, but these tliree are the most important. 

For practical calculations, the microcanonical ensemble is not as useful as other ensembles corresponding to more connnonly occurring experimental situations. Such equilibrium ensembles are considered next. 

One of the flexibilities of eomputer simulation is that it is possible to define the themiodynamie eonditions eorresponding to one of many statistieal ensembles, eaeh of whieh may be most suitable for the purpose of the study. A knowledge of the underlying statistieal meehanies is essential in the design of eorreet simulation methods, and in the analysis of simulation results. Flere we deseribe two of the most eommoir statistieal ensembles, but examples of the use of other ensembles will appear later in the ehapter. 

As we will see further in the book, almost all methods for calculating free energies in chemical and biological problems by means of computer simulations of equilibrium systems rely on one of the three approaches that we have just outlined, or on their possible combination. These methods can be applied not only in the context of the canonical ensemble, but also in other ensembles. As will be discussed in Chap. 5, AA can be also estimated from nonequilibrium simulations, to such extent that FEP and TI methods can be considered as limiting cases of this approach. 

The method is based on the following expression for constant-AVT simulations (modified expressions are available in other ensembles [1])... 

Similar schemes to the above can be used in molecular dynamics simulations in other ensembles such as those at constant temperature or constant pressure (see Frenkel and Smit, and Allen and Tildesley (Further reading)). A molecular dynamics simulation is computationally much more intensive than an energy minimization. Typically with modern computers the real time sampled in a simulation run for large cells is of the order of nanoseconds (106 time steps). Dynamical processes operating on longer time-scales will thus not be revealed. 

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

We will delay a more detailed discussion of ensemble thermodynamics until Chapter 10 indeed, in this chapter we will make use of ensembles designed to render the operative equations as transparent as possible without much discussion of extensions to other ensembles. The point to be re-emphasized here is that the vast majority of experimental techniques measure molecular properties as averages - either time averages or ensemble averages or, most typically, both. Thus, we seek computational techniques capable of accurately reproducing these aspects of molecular behavior. In this chapter, we will consider Monte Carlo (MC) and molecular dynamics (MD) techniques for the simulation of real systems. Prior to discussing the details of computational algorithms, however, we need to briefly review some basic concepts from statistical mechanics. 

Universality holds if a distribution applies not only to the Gaussian ensembles but also to the other ensembles based on the different orthogonal polynomials, such as the Legendre ensembles, within each of the three Dyson universality classes OE, UE, and SE [73],... 

Statistical thermodynamics gives us the recipes to perform this average. The most appropriate Gibbsian ensemble for our problem is the canonical one (namely the isochoric-isothermal ensemble N,V,T). We remark, in passing, that other ensembles such as the grand canonical one have to be selected for other solvation problems). To determine the partition function necessary to compute the thermodynamic properties of the system, and in particular the solvation energy of M which we are now interested in, of a computer simulation is necessary [1],... 

In proving statements (XIII) and (XUIa), we use in the definition of canonical ensemble from other ensembles with the supplementary condition mentioned in (XIV), it was necessary to construct a in a special way (Eq. 63). 

This ensemble, seemingly so "general," has not seen general use. Other ensembles can be invented, with other variables held constant, but they have... 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

An ensemble may be characterized by parameters which are fixed, and those which can be derived from the simulation data, as shown in Table 16.1. Ensembles generated by MC techniques are naturally of the constant NVT type, while MD methods naturally generate a constant NVE ensemble. Both MC and MD methods, however, may be modified to simulate other ensembles, as described in Section 16.2. Of special... 

In the next paragraph, variations of the energy with shape functions are placed in a broader context, dropping the constraint of constant N and looking for analogues of the 8E/8a(r) functional derivatives to other ensembles. 

Other ensembles however exist focusing on [38] other pairs of variables which can be used to describe the passing from one ground state of a system to another as can happen, for example, in a chemical reaction. Through the appropriate Legendre transformation one obtains... 

MC and MD techniques for generating configurations in the other ensembles (with other thermodynamic variables held constant) have been developed and are described in AUen and Tildesley [7] and Frenkel and Smit [8]. 

Another way to view MD simulation is as a technique to probe the atomic positions and momenta that are available to a molecular system under certain conditions. In other words, MD is a statistical mechanics method that can be used to obtain a set of configurations distributed according to a certain statistical ensemble. The natural ensemble for MD simulation is the microcanonical ensemble, where the total energy E, volume V, and amount of particles N (NVE) are constant. Modifications of the integration algorithm also allow for the sampling of other ensembles, such as the canonical ensemble (NVT) with constant temperature... 

In the previous section, we introduced the MDF in the canonical ensemble, i.e., the MDF in a closed system with fixed values of T, V, N. Similarly, one can define the MDF in any other ensemble, such as the T, P, N ensemble. Of particular interest, for this book, are the MDFs in the grand canonical ensemble, i.e., the MDF pertaining to an open system characterized by the variables T, V, /i. The fundamental probability in the grand canonical ensemble is... 

In this section, we used the T, V, N ensemble to obtain relation (2.127). A similar relation can be obtained for any other ensemble. Of particular importance is the analog of (2.127) in the T, P, N ensemble. It has the same form but the events occur in a T, P, N system and instead of the Helmholtz energy change, we need to use the Gibbs energy change. 

We have used the variables T, P, N to define, the chemical potential and the solvation process. These are the most common variables used in practice. However, one can define solvation quantities in any other ensemble. Sometimes it is more convenient in theoretical work to use the T, V, N ensemble. 

Finally, we note that for convenience, we have used the T, V, N ensemble in this section. Similar results may be obtained in any other ensemble as well. 

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